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This book introduces a new class of non-associative algebras related to certain exceptional algebraic groups and their associated buildings. Richard Weiss develops a theory of these "quadrangular algebras" that opens the first purely algebraic approach to the exceptional Moufang quadrangles. These quadrangles include both those that arise as the spherical buildings associated to groups of type E6, E7, and E8 as well as the exotic quadrangles "of type F4" discovered earlier by Weiss. Based on their relationship to exceptional algebraic groups, quadrangular algebras belong in a series together with alternative and Jordan division algebras. Formally, the notion of a quadrangular algebra is derived from the notion of a pseudo-quadratic space (introduced by Jacques Tits in the study of classical groups) over a quaternion division ring. This book contains the complete classification of quadrangular algebras starting from first principles. It also shows how this classification can be made to yield the classification of exceptional Moufang quadrangles as a consequence. The book closes with a chapter on isotopes and the structure group of a quadrangular algebra. Quadrangular Algebras is intended for graduate students of mathematics as well as specialists in buildings, exceptional algebraic groups, and related algebraic structures including Jordan algebras and the algebraic theory of quadratic forms.

Forms, Quadratic. --- Algebra. --- Mathematics --- Mathematical analysis --- Quadratic forms --- Diophantine analysis --- Forms, Binary --- Number theory --- Algebra over a field. --- Algebraic group. --- Associative property. --- Axiom. --- Classical group. --- Clifford algebra. --- Commutator. --- Defective matrix. --- Division algebra. --- Fiber bundle. --- Geometry. --- Isotropic quadratic form. --- Jacques Tits. --- Jordan algebra. --- Moufang. --- Non-associative algebra. --- Polygon. --- Precalculus. --- Projective plane. --- Quadratic form. --- Simple Lie group. --- Subgroup. --- Theorem. --- Vector space.

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This book presents a new result in 3-dimensional topology. It is well known that any closed oriented 3-manifold can be obtained by surgery on a framed link in S 3. In Global Surgery Formula for the Casson-Walker Invariant, a function F of framed links in S 3 is described, and it is proven that F consistently defines an invariant, lamda (l), of closed oriented 3-manifolds. l is then expressed in terms of previously known invariants of 3-manifolds. For integral homology spheres, l is the invariant introduced by Casson in 1985, which allowed him to solve old and famous questions in 3-dimensional topology. l becomes simpler as the first Betti number increases. As an explicit function of Alexander polynomials and surgery coefficients of framed links, the function F extends in a natural way to framed links in rational homology spheres. It is proven that F describes the variation of l under any surgery starting from a rational homology sphere. Thus F yields a global surgery formula for the Casson invariant.

Chirurgie (Topologie) --- Drie-menigvuldigheden (Topologie) --- Heelkunde (Topologie) --- Surgery (Topology) --- Three-manifolds (Topology) --- Trois-variétés (Topologie) --- 3-manifolds (Topology) --- Manifolds, Three dimensional (Topology) --- Three-dimensional manifolds (Topology) --- Low-dimensional topology --- Topological manifolds --- Differential topology --- Homotopy equivalences --- Manifolds (Mathematics) --- Topology --- 515.16 --- 515.16 Topology of manifolds --- Topology of manifolds --- 3-manifold. --- Addition. --- Alexander polynomial. --- Ambient isotopy. --- Betti number. --- Casson invariant. --- Change of basis. --- Change of variables. --- Cobordism. --- Coefficient. --- Combination. --- Combinatorics. --- Computation. --- Conjugacy class. --- Connected component (graph theory). --- Connected space. --- Connected sum. --- Cup product. --- Determinant. --- Diagram (category theory). --- Disk (mathematics). --- Empty set. --- Exterior (topology). --- Fiber bundle. --- Fibration. --- Function (mathematics). --- Fundamental group. --- Homeomorphism. --- Homology (mathematics). --- Homology sphere. --- Homotopy sphere. --- Indeterminate (variable). --- Integer. --- Klein bottle. --- Knot theory. --- Manifold. --- Morphism. --- Notation. --- Orientability. --- Permutation. --- Polynomial. --- Prime number. --- Projective plane. --- Scientific notation. --- Seifert surface. --- Sequence. --- Summation. --- Symmetrization. --- Taylor series. --- Theorem. --- Topology. --- Tubular neighborhood. --- Unlink.

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This book gives a new foundation for the theory of links in 3-space modeled on the modern developmentby Jaco, Shalen, Johannson, Thurston et al. of the theory of 3-manifolds. The basic construction is a method of obtaining any link by "splicing" links of the simplest kinds, namely those whose exteriors are Seifert fibered or hyperbolic. This approach to link theory is particularly attractive since most invariants of links are additive under splicing.Specially distinguished from this viewpoint is the class of links, none of whose splice components is hyperbolic. It includes all links constructed by cabling and connected sums, in particular all links of singularities of complex plane curves. One of the main contributions of this monograph is the calculation of invariants of these classes of links, such as the Alexander polynomials, monodromy, and Seifert forms.

Algebraic geometry --- Differential geometry. Global analysis --- Link theory. --- Curves, Plane. --- SINGULARITIES (Mathematics) --- Curves, Plane --- Invariants --- Link theory --- Singularities (Mathematics) --- Geometry, Algebraic --- Low-dimensional topology --- Piecewise linear topology --- Higher plane curves --- Plane curves --- Invariants. --- 3-sphere. --- Alexander Grothendieck. --- Alexander polynomial. --- Algebraic curve. --- Algebraic equation. --- Algebraic geometry. --- Algebraic surface. --- Algorithm. --- Ambient space. --- Analytic function. --- Approximation. --- Big O notation. --- Call graph. --- Cartesian coordinate system. --- Characteristic polynomial. --- Closed-form expression. --- Cohomology. --- Computation. --- Conjecture. --- Connected sum. --- Contradiction. --- Coprime integers. --- Corollary. --- Curve. --- Cyclic group. --- Determinant. --- Diagram (category theory). --- Diffeomorphism. --- Dimension. --- Disjoint union. --- Eigenvalues and eigenvectors. --- Equation. --- Equivalence class. --- Euler number. --- Existential quantification. --- Exterior (topology). --- Fiber bundle. --- Fibration. --- Foliation. --- Fundamental group. --- Geometry. --- Graph (discrete mathematics). --- Ground field. --- Homeomorphism. --- Homology sphere. --- Identity matrix. --- Integer matrix. --- Intersection form (4-manifold). --- Isolated point. --- Isolated singularity. --- Jordan normal form. --- Knot theory. --- Mathematical induction. --- Monodromy matrix. --- Monodromy. --- N-sphere. --- Natural transformation. --- Newton polygon. --- Newton's method. --- Normal (geometry). --- Notation. --- Pairwise. --- Parametrization. --- Plane curve. --- Polynomial. --- Power series. --- Projective plane. --- Puiseux series. --- Quantity. --- Rational function. --- Resolution of singularities. --- Riemann sphere. --- Riemann surface. --- Root of unity. --- Scientific notation. --- Seifert surface. --- Set (mathematics). --- Sign (mathematics). --- Solid torus. --- Special case. --- Stereographic projection. --- Submanifold. --- Summation. --- Theorem. --- Three-dimensional space (mathematics). --- Topology. --- Torus knot. --- Torus. --- Tubular neighborhood. --- Unit circle. --- Unit vector. --- Unknot. --- Variable (mathematics).

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This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Curves, Algebraic. --- Finite fields (Algebra) --- Modular fields (Algebra) --- Algebra, Abstract --- Algebraic fields --- Galois theory --- Modules (Algebra) --- Algebraic curves --- Algebraic varieties --- Abelian group. --- Abelian variety. --- Affine plane. --- Affine space. --- Affine variety. --- Algebraic closure. --- Algebraic curve. --- Algebraic equation. --- Algebraic extension. --- Algebraic function. --- Algebraic geometry. --- Algebraic integer. --- Algebraic number field. --- Algebraic number theory. --- Algebraic number. --- Algebraic variety. --- Algebraically closed field. --- Applied mathematics. --- Automorphism. --- Birational invariant. --- Characteristic exponent. --- Classification theorem. --- Clifford's theorem. --- Combinatorics. --- Complex number. --- Computation. --- Cyclic group. --- Cyclotomic polynomial. --- Degeneracy (mathematics). --- Degenerate conic. --- Divisor (algebraic geometry). --- Divisor. --- Dual curve. --- Dual space. --- Elliptic curve. --- Equation. --- Fermat curve. --- Finite field. --- Finite geometry. --- Finite group. --- Formal power series. --- Function (mathematics). --- Function field. --- Fundamental theorem. --- Galois extension. --- Galois theory. --- Gauss map. --- General position. --- Generic point. --- Geometry. --- Homogeneous polynomial. --- Hurwitz's theorem. --- Hyperelliptic curve. --- Hyperplane. --- Identity matrix. --- Inequality (mathematics). --- Intersection number (graph theory). --- Intersection number. --- J-invariant. --- Line at infinity. --- Linear algebra. --- Linear map. --- Mathematical induction. --- Mathematics. --- Menelaus' theorem. --- Modular curve. --- Natural number. --- Number theory. --- Parity (mathematics). --- Permutation group. --- Plane curve. --- Point at infinity. --- Polar curve. --- Polygon. --- Polynomial. --- Power series. --- Prime number. --- Projective plane. --- Projective space. --- Quadratic transformation. --- Quadric. --- Resolution of singularities. --- Riemann hypothesis. --- Scalar multiplication. --- Scientific notation. --- Separable extension. --- Separable polynomial. --- Sign (mathematics). --- Singular point of a curve. --- Special case. --- Subgroup. --- Sylow theorems. --- System of linear equations. --- Tangent. --- Theorem. --- Transcendence degree. --- Upper and lower bounds. --- Valuation ring. --- Variable (mathematics). --- Vector space.

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One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important book on the subject for more than forty years. Morse theory was developed in the 1920s by mathematician Marston Morse. (Morse was on the faculty of the Institute for Advanced Study, and Princeton published his Topological Methods in the Theory of Functions of a Complex Variable in the Annals of Mathematics Studies series in 1947.) One classical application of Morse theory includes the attempt to understand, with only limited information, the large-scale structure of an object. This kind of problem occurs in mathematical physics, dynamic systems, and mechanical engineering. Morse theory has received much attention in the last two decades as a result of a famous paper in which theoretical physicist Edward Witten relates Morse theory to quantum field theory. Milnor was awarded the Fields Medal (the mathematical equivalent of a Nobel Prize) in 1962 for his work in differential topology. He has since received the National Medal of Science (1967) and the Steele Prize from the American Mathematical Society twice (1982 and 2004) in recognition of his explanations of mathematical concepts across a wide range of scienti.c disciplines. The citation reads, "The phrase sublime elegance is rarely associated with mathematical exposition, but it applies to all of Milnor's writings. Reading his books, one is struck with the ease with which the subject is unfolding and it only becomes apparent after re.ection that this ease is the mark of a master.? Milnor has published five books with Princeton University Press.

Homotopy theory. --- Geometry, Differential. --- Affine connection. --- Banach algebra. --- Betti number. --- Bott periodicity theorem. --- Bounded set. --- Calculus of variations. --- Cauchy sequence. --- Characteristic class. --- Clifford algebra. --- Compact space. --- Complex number. --- Conjugate points. --- Coordinate system. --- Corollary. --- Covariant derivative. --- Covering space. --- Critical point (mathematics). --- Curvature. --- Cyclic group. --- Derivative. --- Diagram (category theory). --- Diffeomorphism. --- Differentiable function. --- Differentiable manifold. --- Differential geometry. --- Differential structure. --- Differential topology. --- Dimension (vector space). --- Dirichlet problem. --- Elementary proof. --- Euclidean space. --- Euler characteristic. --- Exact sequence. --- Exponentiation. --- First variation. --- Function (mathematics). --- Fundamental lemma (Langlands program). --- Fundamental theorem. --- General position. --- Geometry. --- Great circle. --- Hessian matrix. --- Hilbert space. --- Homomorphism. --- Homotopy group. --- Homotopy. --- Implicit function theorem. --- Inclusion map. --- Infimum and supremum. --- Jacobi field. --- Lie algebra. --- Lie group. --- Line segment. --- Linear equation. --- Linear map. --- Loop space. --- Manifold. --- Mathematical induction. --- Metric connection. --- Metric space. --- Morse theory. --- N-sphere. --- Order of approximation. --- Orthogonal group. --- Orthogonal transformation. --- Paraboloid. --- Path space. --- Piecewise. --- Projective plane. --- Real number. --- Retract. --- Ricci curvature. --- Riemannian geometry. --- Riemannian manifold. --- Sard's theorem. --- Second fundamental form. --- Sectional curvature. --- Sequence. --- Simply connected space. --- Skew-Hermitian matrix. --- Smoothness. --- Special unitary group. --- Square-integrable function. --- Subgroup. --- Submanifold. --- Subset. --- Symmetric space. --- Tangent space. --- Tangent vector. --- Theorem. --- Topological group. --- Topological space. --- Topology. --- Torus. --- Unit sphere. --- Unit vector. --- Unitary group. --- Vector bundle. --- Vector field. --- Vector space.